MAXIMUM ERROR IN SOME MINERALOGIC
COMPUTATIONS
H. W. FarnsarnwaNo C. W. Surrrano,
gy, Cambridge,M ass.
M ossochus
et'tsI nstituteoJ T echnolo
CoNrnNrs
ABsrRAcr.
IutnooucuoN.
DnrnnurmlrroN
ol DExsrw.
Suspnrsror METrroD.
Hvnnosrmrc METrroD
Jolly balance.
Berman balance
Pvcxourtnn
METrroD
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METrroD.
Coupenarrvr AccuRAcy
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oprrc lNor,n METrroD. . .
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coup^nerrvo AccuRAcy.
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ol Oprrc Alrcr,n.
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rNDEx METHoD.
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ABsrRAcr
The notable lack of quantitative information regarding errors in many mineralogic
computations has prompted the writers to construct a series of diagrams from which such
information can be readily obtained. These include various determinative methods for
density, refractive index, birefringence, and optic angle. Assuming a skilled operator,
maximum values of the error in single observations are assigned to each procedure, so that
the diagrams represent "worst" values of error, rather than a "probablet' error as determined from a series of observations. Comparison of the various procedures is based on this
"maximum expected" error.
o/J
674
H, W, FAIRBAIRN AND C. W, SHEPPARD
INr:nopucrroN
It is a truism that all physical measurements are approximations
which approach, in varying degree, an unattainable absolute value.
As a result it is essential that the degree of approximation, or amount
of fractional deviation, be adequately known. In the field of mineralogy
everyone knows about this, but few have consiStently done anything
about it. For various reasonsthe subject as a whole has been treated in
the manner of a poor relation, with much overstatement and someunderstatement of the fractional deviations obtained (Hey, 1933). The
present paper is an attempt to improve this situation.
Deviations (also called uncertainties or errors) are classified as systematic (periodic) or random (accidental). Systematic errors are inherent in every experimental investigation (e.g. temperature variations
in density and refractometric work) and can be controlled. This control is in some casesfine enough so that this type of error is negligible
compared with the random error. Random derliations arise through
reading of scales, adjustment of extinction positions, etc. For a single
observation the uncertainty o{ the measurement depends in part on the
skill of the observer, in part on the eharacter of the apparatus. If repeated observations are made, the random error may be expressedquantitatively by calculating a "probable error." As there is a 50/e probability of a measurement difiering from the true value by more than this
amount, the "probable error" of an observation may give a false impression of accuracy. In the present discussion "maximum expected
error" is considered only-i.e., it represents the probable worst value
which a skilled operator would obtain from a single measurement with
a given apparatus. If it is known in advance how large this maximum
expected error is likely to be and if this value, added to the known systematic error, gives a total uncertainty which can be tolerated for the
work in hand, the labor of computing a standard "probable error" may
be avoided. This maximum error is of course not a precise quantity, but
the numerical values used in constructing the diagrams given here are
believed to be large enough so that the curves give fractional deviations
which would be exceededonly in rare instances.
The general method followed is (1) differentiation of the basic formula
with respect to all pertinent variables, (2) assumption of maximum expected values for the difierentials, (3) solution of the difierential equation
to determine the maximum expectedrandom error, (4) calculation of the
systematic error, (5) addition of random and systematic errors, (6) con. struction of a diagram.
It is assumedthat the reader is familiar with the determinative methods under scrutiny and has acquired enough manipulative skill to avoid
random errors appreciably greater than those shown in the diagrams.
MAXIMUM
ERROR IN MINERAI.OGIC COMPUTATIONS
o/5
In caseswhere the observer believes that the uncertainty in any of his
own measurements differs markedly from the values used here, an adjusted diagram can usually be made with little labor by referring to the
proper differential equation.
DnrnnurNauoN or, DnNsrrv
Four commonly used methods of density determination are considered and their relative accuraciescompared. The suspension,hydrostatic,
and pycnometer procedures give relative density, or "specific gravity,"l
whereasthe r-radiation method measuresdensity directly, without use of
a reference fluid such as water or toluene.
SuspeNsroN MErr{oD. Although this method does not involvp any
computation and is limited in its application, its potential accuracy is
worthy of note, especially where Clerici's solution is used (Jahns, 1939).
The random error in density inherent in Jahns'refractive index-density
curve is given as .01. A second random error arises from reading the
Abb6 refractometer, and may be assumed to be .0005. Using the slope of
this curve to obtain the ratio of density and index change (dD/dn), the
uncertainty in density is found to be .00044, negligible in comparison
with the first random error. The total random error of .0104 is converted
to a fractional deviation dD/D, this value being plotted against D in
Fig.2 (inset, top left). It is assumedthat the systematic error arising
from temperature variations can be adequately controlled.
Extension of the density range through use of a glass float (of known
weight and density) attached to the mineral fragment can be achieved,
but with relative loss in accuracy. This case is not consideredhere (see
Bannister and Hey, 1937).
Hvnnosrerrc METHoD.A number of rnethods of this tvpe have been
described,of which two are discussedhere.
Jolly Balance.The expressionfor density is
WeDw
lJ:wo-w*'
where Wr:weight of substancein air.
Ww:weight of substancein water.
Dw:density of water at room temperature relative to D*:1
at 40C.
t Although widely used,
"specific gravity" is terminologically incorrect as a syronym
for relative density, and is physically incorrect as a synonym for density determined by
the r-radiation method. "Gravity" implies the weight of a body, i.e., the earth's athaction
for it, which is not an intrinsic property, whereas by definition density is the mass of a
body per unit volume, the mass being an intrinsic and invariable property.
676
H. W. FAIRBAIRN AND C. W. SHEPPARD
The fractional deviation is given by
dD
D
WedWw * WwdWe
dDw
D*'
Wl(Wo - Ww)
where dWw and dWe are the uncertainties in determination of Wa and
Ww, respectively, and dDw is the uncertainty in density of water at arry
temperature. As dWe and dWw are obviousiy equal they may be replaced by dW. Furthermore, the uncertainty in Dw is relatively negligible so that, finally,
dD
dW(Wr * Ww)
D
W,r(Wr - Ww)
which represents the {ractional deviation arising from the random errors
of reading the scale. The obvious source of systematic error would be
temperature variations which would affect the density of the water.
Ilowever, it is found that, between 15o and 25oC. (room temperature
range), this error may be neglected.
Figure t has been prepared from data obtained with the current apparatus ofiered by the Central Scientific Company. Two springs are
provided to extend the range of the balance. The uncertainty of any individual reading of the scale should not exceedone-quarter of a division.
In order to compare directly with errors found in the other weighing
methods, the springs were calibrated in grams, giving dW:.005 gm. for
the light spring, and dW: .025gm. for the heavy spring.2This conversion
is of coursenot ordinarily necessary.Per cent fractional deviation is plotted against density for each spring, using four values of We known to be
within the range of the scale.The sensitivity of each spring is considered
constant within the range selected. It is seen from Fig. I that least accuracy (greatest fractional deviation) is obtained where We is low and D
is high, and, conversely, that maximum accuracy is to be expected from
the combination of high W1 and low D. Furthermore, the variation in
dD/D with D is much lessfor high We than for low Wr. These observations hold also, in varying degree, for the other balance methods discussed(Figs. 2 and 3).
Berman Balance.As with the Jolly balance the Iractional deviation rs
g?: dw(we* ww) - T - td D r
D
We(We - Ww)
Dr
where dD1 and Dr refer to toluene instead of water. At any given tem2 Figure 1 will probably require some
adjustment for Jolly balances of difierent make.
MAXIMUM
ERROR IN MINERALOGIC COMPUTATIONS
677
perature there is an uncertainty dDr:.0005 in the density of toluene
(Timmermans, 1912).The random error dW of a singlescalereading on
2
.9
=
E
,(L
?
LI
!
Frc. L lotl'y balance. Variation in accuracy for difterent densities' Random error computed for Wa values of 1,2,3,4 gm. (using light spring) and 10, 20, 30, 40 gm. (using heavy
spring). Based on an error of .25 division in reading the scale.
o1
3
o'2
-2\/
D
Fre.2. Berman balance and' (inset, top LeJt) Clerici solution rnelhod (Johns). Random
error for Berman balance computed for Wa values of 5, 15, 20, 25, 50 mg. Based on dW
:.01 mg. Full lines (15,20,25 mg.) cover the range of maximum sensitivity of the balance
as specified by the manufacturer. Random error for Clerici solution method based on experimental data given by Jahns (1939).
678
H. W, FAIRBAIRN AND C. W, SHEPPARD
the balance should not exceed .01 mg. The systematic error arising from
determinations made at various temperatures is relatively negligible in
the range 15o-25oc. Surface tension deviation may also be neglected
where the suspension wire diameter is a small fraction of millimeter.
Figure 2 is therefore constructed directly from the equation given above.
Optimum conditions of accuracy are realized only for We values from
15 to 25 mg., as specifiedby the manufacturer.
Where a determination is made with several small fragments instead
of with a larger, single piece, accuracy will probably be lowered because
of less perfect wetting. Unlike the pycnometric method, this systematic
error can not be eliminated.
PvcNol.rornR MErHoD.Density by the pycnometer procedure is given
by
D:
(W, - W')Dr
Wz-W*DrV
where W1:ra'eight of pycnometer
W2: weight of pycnometer*powder
WB:weight of pycnometerf powder{toluene
Dr:density of toluene
V : volume of pycnometer.
If a silica-glasspycnometer is used, the uncertainty in V may be reduced
by careful calibration to a relatively negligible value (Ellsworth, 1928).
In the following expressionfor fractional deviation V is therefore considered constant
dD
2dW
-T--
D
Wz-Wr
dDr
Dr
The three uncertainties dW1, dW2, dW3 used in the first step of the differentiation are equal and have been replaced by dW in the final condensedexpressionabove.sAs the uncertainty in weighing on the ordinary
3 As the writers have not succeeded in duplicating the
expressions derived by one or
trvo other authors for the pycnometric method, the derivation used here is given in detail.
D:
(Wz - Wr)Dr
W z - W s * DrV
Taking logarithms of each side,
logD:logDr*
Difierentiating,
dD
D
log (Wz - Wr) - log (Wz - W3 + DrV)"
dDr . dW, - dW1
Dt
Wr-W,
dWz-dWr+VdD.r
Wz-We*DtV
MAXIMUM
ERROR IN MINERAI.OGIC COMPUTATIONS
chemical balance should not exceed .0001 gm., this value is assignedto
dW in constructing Fig. 3. dDr:.0005 as in Fig' 2' Room temperature
variations produce relatively negligible systematic errors. The 10 cc.
bottle described by Ellsworth (1928) is used to show variations in accuracy for powder samplesweighing l, 2, 5, and 10 gm. ; the 0. 1 cc. bottle
describedby Winchell (1938) is used similarly for powder weights of 0.1,
0.2, and 0.5 gm.
Every precaution must be taken to wet the powder thoroughly, as
failure to remove all the imprisoned air will lead to deviations much
greater than the rand.omerror from weighing. Ksanda and Merwin (1939)
have described an excellent technique of this kind.
Figure 3 shows that bottles of large capacity give smaller fractional
deviations over the usual range of powder weights than do bottles of
small volume, assuming equivalent conditions of weighing. This becomes
obvious from inspection of the differential equation' Increase in V reducesthe dDr/Dr term, and increasein Wz-Wr (weight of powder in air)
reduces the dW term. Ofisetting this apparent superiority of the larger
bottle, however, is the fact that its greater bulk of powder increasesthe
Collecting the dWr, dWz, dWa, and dDr terms separately,
_u*,(*,:w,)
*:'"'(uf
+ aw'(*r- 'r, + dwr(
Wz-Ws*DtV
)
For determination of m,arimum error all the bracketed expressions must have the same sign.
The dWr term is therefore made positive. (The minus signs wi.thin the dDr and dWz
brackets must be left unchanged, as Da and W2 occur in both numerator and denominator
of the original density equation. Errors in their measurement would therefore be expected
to compensate each other partially, as shown by the minus signs in the diflerential equawe replace each by dW. Thus
tion.) Furthermore, as dWr:dWz:dWe,
dD
-:d
D
--- - w r
Wr
w;+
I
Dt\r)l
lVs f DrV) * (W' - W3 + DrV) - (WI - W1) + (W,
(Wz-Wr)(Wz-W'+DrV)
constanttheterm
If V is not considered
ttr
-+=
#5*"]
mustbeaddedtotheabove'
680
H. W. FAIRBAIRN AND C. W. SHEPPARD
difficulty of removing the last traces of trapped air. Homogeneity in composition of the sample is also more difficult to attain with a large sample.
It is probable therefore that in practice the accuracy of the two bottles
will be about the same.
In the micropycnometric procedure described by Bannister and Hey
(1937) only 15 to 25 mg. of sample are required, with a reported accuracy
oI 0.5/6. As part of this error stems from an uncertainty in the volume
of liquid used, a greater error than that of Fig. 3 would be expected.
pycnomr,r,
Fte.3.
*l,roo.u".t",a"; ':"i,
a,
j,u","r'r'l"nsities.
Random
error
for 10 cc. bottle (Ellsworth type) and.1 cc. bottle (Winchell type). FuIl lines refer to the
10 cc. bottle for mineral weights (in air) of 7,2,5,l0 gm.; broken lines refer to the .1 cc.
bottleformineralweights(inair) of .1,.2,.5 gm. Based on dW:.0001 gm. and dDi:
.0005.
However, uncertainties regarding the homogeneity and wetting of the
powder may be better controlled in the Bannister-Hey method than in
the others, and too much emphasis should not be placed therefore on the
purely mathematical aspectsof the case.
X-nanrarron MErHoD.Density by this method is expressedby
o:
MZ
ou
where M: molecular weight of the material
Z:number of formula-units in a unit cell
V:volume of unit cell
A: Avogadro's number (6.02X 1O'?).
ERROR IN MINERALOGIC COMPATATIONS
MAXIMUM
681
As V is the single variable,a
AD
AV
In a cubic crystal, oo:d(h2*k2+12)rt2
where o6:lsngth of side of unit cell,
d:interplanar spacing of planes with indices lftl'
Aao
Ad
2,0
d
Therefore
aV
But -V
A(ao3)_
ao3
3(aao)_
o'o
3ad.
d
From Bragg's equation d:tr/2 sinO,
where tr:wave Iength of r-radiation used'
0: diffraction angle.
The fractional deviation of d is ad/d: -cot 0L0. Expressing the fractional deviation in density in terms of the diffraction angle, AD/D:3
cot 0A0(for cubic crystals only).
Application of this equation is confined here to powder photography'
The eiror in read.inga millimeter scaleplaced on a line of the film should
not exceed .25 mm. for any single measurement. Therefore the uncertainty in the distance between such a line and its mate on the other side
of the line of zero difiraction is 0.5 mm. As this total distance in a camera
oi 57.3 mm. diameter is equivalentto 40,the error in d is .125o.That is,
A0:.125/57.3:.00128 radians. The curve R for random error in Fig' 4
is based on this value.
The four systematic errors inherent in *-ray crystal diffraction work
-absorption of radiation by the specimen,eccentricity of the samplewith
respect io the camera, film shrinkage after development, and uncertainty
in the camera radius-must each be consideredon its merits. As details
are given by Buerger (1942) for such corrections the matter is taken up
onlybriefly here. use of a two-hole film (Straumanis method) permits the
camera radius and film shrinkage errors to be combined, and for a carefully constructed camera and film dried at room temperature, the error
from this source is relatively negligible. It does not appear in Fig' 4'
The eccentricity error is small but appreciable. Its curve (E in Fig' 4)
is based on a maximum expected displacement of the sample relative to
the true center of the camera of .025 mm. The equation used is given by
Buerger (1942,p.413). The deviation may be positive or negative'
a To avoid conlusion with the interplanar spacing symbol d, A replaces here the customary difierential symbol.
682
E. W. FAIRBAIRN AND C. W. SHEPPARD
The absorption error (Buerger, 1942, p. 414) is based on complete
absorption, a specimen diameter of .25 mm., a pin-hole-specimen distance of 10.6 mm., and camera diameter 57.3 mm. Its curve (A in Fig.
4) shows an order of magnitude for AD/D similar to the curve R, and of
the samesign.
The total value of the maximum expected error is shown by curve T
and has been drawn using positive values for the individual errors. As
the random error may be opposite in sign to the absorption error, and is
50'
60'
e
70'
80'
90'
Fte .4. X-ray methoil.Yariationin accuracyfor difierent difiraction angles. Random error
(curve R) based on an uncertainty of .25 mm. in reading the position of a diffraction line.
Absorption error (curve A) from data in Buerge r $9a\. Eccentricity error (curve E) likewise from data in Buerger (1942). Curve T shows total error where R, A, E are assumed to
have the same sign.
of the same order of magnitude, the total error may, however, be very
small. Curve T represents the worst possible case. The diagram does
not include values smaller than 50o, as the accuracy decreasesvery rapidly in this range.
CoupanerrvE AccuRAcy. Excluding the x-ray method, comparison of
the remaining procedures shows that the pycnometer method has the
greatest inherent accuracy, the Jolly balance method the least. Choice
of method is dictated by the nature of the material and the degreeof accuracy required. For most work the accuracy range of the Berman balance is adequate, provided that a single fragment of suitable weight can
be obtained.
Direct comparison of these weighing methods with the *-radiation
MAXIMUM
ERROR IN MINERAINGIC
COMPUTATIONS
683
method is not possible, as the latter gives the density of the unit cell
whereasthe former give values for aggregatesof unit cells.The individual
cells in these aggregatesare rarely, if ever, packed together perfectly,
with the result that lower density values are obtained than from the *-ray
procedure. If it is desired to contrast the two density values for a given
material, a weighing method should be selectedwhich gives an accuracy
comparable with that indicated by the 0 value chosen in Fig. 4' In general, the Berman balance would satisfy this requirement.
DBrnnurNarloN or RBlnacuvn
INoBx
As this paper is concerned with errors in computation only, the accuracy of direct-reading instruments will not be discussed.In the caseof
refractometers the accuracy of the widely used Abb6 type has recently
been reviewed by Tilton (1942) in an informative paper. It need only be
stated here that the ordinary Abb6 used in mineralogical laboratories
reads correctly to *.0002 in its lower range, but is considerably less
accurate in its upper range. Other direct-reading re{ractometers (Fisher,
Nichols, etc.) are usually reliable to *.001.
Mruruuu DEvrArroN MErHoD.In the field of precise refractometry the
minimum deviation method is preferable to any other. Where moderate
precision is required, as in standardization of oils or melts for use in the
immersion method of refractive index determination, it affords the only
universal method for the entire range of indices. The accuracy of the
method has been discussedby Tilton (1935) who lists 19 sourcesof error
for determinations of the highest precision (+.000001). As mineralogists
seldom require this accuracy for crystal measurement,and never for immersion oils or melts, only the random error of reading prism and minimum deviation angles need be consideredhere.
The index of refraction is given by
. A+D.
Sln -
a
L
.A
SIN -
2
where A:angle of the prism.
D- : minimum deviation angle.
sinD^f2 cosec2
Difierentiating partially with respect to A, dn: -I/2
A/2 d A, and partially with respect to D^, dn--1/2 (cos D*/2 cot h/2
684
H. W. FAIRBAIRN AND C. W, SHEPPARD
-sin D*/2) dD-. As it is probablesthat dA:dD*,
dr, so that the total random error is
we replaceeach by
A D^ j
z(dn): f[.o*r.1 .o, +
cosec2rin3=lor.
2L
2
2
2----21-"
rn constructing Fig. 5 the error in the prism and minimum deviation
angles is assumedto be *1/, so that dr:.000291 radians. The second
term in the difierential equation is considered positive in order to give
Ftc. 5. Minimurn d'ettiationmethod,.l{aiation of random error in z for a series of z values.
Computations shown for prism angles of 25', 30",40o, 50o, 60", with the
/a accuracy given
for each' Steep curve transecting the prism curves is the locus of limitingvaluesof prism
angles for determination of a given refractive index. Diagram based on an uncertainty of
* 1r in determination of prism and minimum deviation ansles.
dn its maximum expected value.6 Most single determinations wilr have
greater accuracy than that indicated by the graph. In working with crystal prisms the deviation is likely to be much less than 1,, but with the
ordinary hollow cell used for oil determinations this is unlikely, and for
crude prisms composedof S-Semixtures may be much greater than 1/.
Figure 5 showsthe total maximum expectederror for prisms of varying
6 This may not be true if the prism
angle is determined from readings of the table circle,
and the minimum deviation angle from readings of the telescopecircle. The table angles of
small spectrometers and goniometers tend to be less accurate than telescopeangles. However, if the prism angle is determined by the split-beam method, no table angle need be
rtsed, and except in high-precision work gives very satisfactory results (Guild, 1923).
6 The small numerical range of the
common refractive indices makes it unnecessarv to
plot the variation in fractional deviation for difierent values of n.
MAXIMUM
ERROR IN MINERAI.OGIC COMPUTATIONS
685
angle over a wide range of indices. For a given index greatest accuracy
(least dn/n/) is attained where large prisms are used. Inspection of the
formula
. A+D^ /. A
,?:srn
2 /"t"1
indicatesa maximum value of nlor any given prism angle. The boundary
curve in Fig. 5 showsthis graphically. For example,the limiting refractive
index determinable with a 600 prism angle is 2.0. In practice, since the
14
15
l'6 n
17
18
19
z'e
Ftc.6. Mini'rnum dpvialion rnethodwhere prism angle is 6O". Curve P shows random error
arising from prism angle uncertainty of 1', Curve D14 shows random error arising from
minimum deviation uncertainty of 1'. Upper curve shows the total error where P and D1,a
are assumed to have the sarne sign.
proportion of transmitted light at grazing incidence is greatly reduced
relative to the reflected portion, it is never possibleto proceed to the limit
shown in the diagram.
Figure 6 gives greater detail for a 60o prism. Curve P shows the
variation in prism angle error for various indices; D* shows dn for different indices arising from measurement of the minimum deviation angle. At the limiting index of 2.0 the prism angle error reachesitsmaximum
whereas the minimum deviation error becomeszero. The upper, straight
line representsthe total maximum expected error. If the Gifiord methodT
7 This method (Gifford, 1902) makes use of a closed prism whose angles approximate
60'. Minimum deviation angles are measured acrosseach prism and an a\erage computed.
Angle A may then be assumed equal to 60o without any accurate measurement of the ino
dividual prisms. An uncertainty in A of the order of magnitude of 1 causes an uncertainty
of only *.0001 in z. As there need be no difficulty in maintaining the prism Ceviations at
much less than 1o, the resultant prism angle error using this method becomes negligible.
686
H. W. FAIRBAIRN
AND C. W. SHEPPARD
is used, the error in prism angle becomesnegligible and the total uncertainty is shown by D^. In all caseswhere the demands of this method
can be met, increasedaccuracy will result, particularly for refractive indices approaching 2.0 in magnitude.
Where the goal of accuracyis in the neighborhoodof *.0001, systematic errors become significant only for determinations made on liquids.
Temperature corrections are necessary,particularly for methylene iodide
dn x l05
Frc. 7. Grazing inciilence mathod (Puffrich refractomcter). Variation in dn with n shown
for t'rvo prisms of refractive index as marked. Approximate /s accuracy at curve maxima
is also given. Based on an uncertainty of * f in reading the scale.
and mixtures containing it, and require control to at least 0.10C.if the index error limit of .0001 is to be maintained. In addition the hollow cell
must be constructed of optically flat glass. Suitable material may be seIected according to the method suggestedby Larsen and Berman (1934).
In Figs. 5 and 6 it is assumedthat these systematic errors can be reduced
to negligible proportions and that only random errors remain.
Gnazrxc rNcrDENcE MErHoD. This method is incorporated in the
widely used Pulfrich refractometer. Although Iessused by mineralogists
than by chemists(Gibb, 1942,p.333), and inherently lessaccuratethan
instruments used for precise minimum deviation work, a brief comparison is not out of place. Refractive index is given by
n :
(np2- sin2r)1/2
where no:lefractive index of prism,
r: angle read on divided circle.
Differentiation gives
sin 2r
-[
d":
2(no' -
]u,
sin2 r)1/2
MAXIMUM ERRORIN MINERALOGICCOMPUTATIONS
687
The error in a single reading in the middle range of r values is not likely
to exceed 1/, so that dr:.000291 radians. Figure 7 shows a graphical
solution of the differential equation for two prisms ol n:I.620 and n
:1.752. For each prism there is an r value which gives maximum fractional deviation (.07/6 and .06/6 as indicated). Thus for material of
n:!.6 the maximum expected error would be only about one-half as
great where the lower index prism is used as where the higher one is used.
For high and low r values, however, dr may easily exceedthe value used
in the diagram, so that the accuracy indicated in the lower part
(dn <.00004) may not be realized. Ilowever, even the worst values at the
noses of the curves are as good, or better, than the accuracy to be expected from ordinary four-figure minimum deviation measurements.
UNrvnnsar, srAGEExrRApoLATroNMETHoD,In caseswhere one critical
index alone can be determined in a given oriented optic symmetry plane,
measurement of an intermediate index at a known angle in this plane
makes possible the computation of the other critical index. This method,
first used by Pauly (Johannsen, 1918, p. 266), and' later developed by
Emmons ( 1943,pl. 10) dependson the following equation
n2(nr2sin2 d *
n*2 cos'6) t/'
rlr
where n*:qnknown critical index,
nr : known critical index,
n2: intermediate index,
d:rotation angle used in obtaining n2.
Assuming that the uncertainties in n1 and nr are the same, and using
dn to represent both, the difierential equation is:
d- n. :*
nx
n*'\
n * z c o t 2 g / l1\1
_ _ ) l d " + (/ I _ _ ; ) c o t{ d { .
: T| _1 + - - (
nr'.2
Lnz
l1z'
\nz
nr/ J
\
If dn:.001 and the uncertainty8 in S is 1" (dO:.0174 radians), the
equation appears graphically as in Fig.8. It is assumed that nr)n*'
In Fig. 8 dn* is plotted against (tr-t*) for difierent values of f using
the computations made for nr:1.6. For other values of nr no appreciable difierence in the deviations is obtained. For d:90o inspection
of the differential equation shows that 41*:(n*/nz) dn and since nz
now equalsn*, then dn*: d1 : .001as shown in Fig. 8. The total dn term
remains fairly constant at .001*, most of the variation being shown by
s The uncertainty in f stems from the initial random error in orienting the optic symmetry plane rather than from any inherent defect in the OEW axis of the stage.
688
H, W. FAIRBAIRN AND C. W, SHEPPARD
the d{ term. In general,dn* is least where large anglesare combined with
low partial birefringence. Except for angles of less than 20o, however,
dn* changeslittle where partial birefringence is low.
Frc.8. Ertropol,ation m,ethod..Variation in dn* for difierent partial birefringence (nr
-n*). Shown for three values of
d, where nr:1.6. Based on dn:.001 and an uncertainty
in d of 1'.
td
F=l50
(+)
d6
de
,/
00(
,///
000
H
q-p
z
-)?'
0l
03
04
Frc.9. Optie angl'ernethoil.Yariationin d7 (or da) for difierent values of 7- B @r a-B).
Four selectedvalues of the optic angle are used. Diagram based on p:1.50 and an optically
positive crystal. FulI lines refer to da only; broken lines to d7. The uncertainty in determination of the known refractive indices is.001 and in the optic angle 1o.
MAXIMUM ERRORIN MINERALOGICCOMPUTATIONS
689
If nr(n*, dn* increasesby a few per cent for high n* and birefringence
values; for low values the change is inappreciable. No diagrams for this
case have been constructed.
Oprrc arqcr,nMErHoD.Mertie's nomogram (Mertie, 1942) showing the
relations between a, A, "y,and V, based on ellipsoid geometry, is a necessary prerequisite to the deviation diagrams of this paper. As differentiation of the full equation gives a long, involved expression,the approximate equations
B-ot
sin2V
'Y-ot
sin2 V :!--J
^l-ot
for optically negative substances,
are used here. Differentiation of these with respect to a, 8,7 in turn,
gives
da:
dn *
(0 - t) sin 2VdV
for optically positive substances.
dy:
da:dn*
sina V
@
t) sin 2VdV
sina V
dv:dn*
@-
")
sin 2VdV
for optically negative substances.
cosa V
dF :
dn * (t - a) sin 2VdV for either positive or negative substances,
where dn substitutes for dB and dt, dg and da, da and d7, respectively,
and is assumedequal to .001. The uncertainty in direct measurement of
2V need not exceed + 10if both optic axesare found. If only one axis can
be oriented the uncertainty will be t 2o. For low to moderate refringence
and birefringence
d"(*)
:
dr(-)
and dr(*)
:
do(-).
Figure 9 is based on a constant value of B: 1.59, an optically positive
substanceand dV:.0174 radians (uncertainty in V of *1o). It shows
the variation in da with change in a-8, and the variation in d7 with
change in,y-0, fpr selectedvalues of V. As would be expected, the closer
a (or 7) approachesB the smaller is da (or dl). The rate of increase in
da (or d7) is greater for small V values than for large. The percent fractional deviation (accuracy) follows the same trend. Figure 10 is similar
690
H. W. FAIRBAIRN AND C. W. SHEPPARD
to 9, but is based on B:1.90. The same type of variation persists,but
da and d7 are on the whole several times larger. For p values other than
1.50 and 1.90 interpolation or extrapolation will give a reasonably good
result.
,//
'l
/,.
1%y
/
d
d
,4,
%
,%
't
a-p
,.8
./:
| 90 (+)
006
Frc. 10. Optic angle m,ethoil..Same as Figure 9, but based on p:1'90'
Frc. 11. Opti,c angl,emcthoil.. Variation in dB for different values of total birefringence
(7-a). Four ielected values of the optic angle are used. Based on uncertainties of .001 in
determination of the known refractive indices, and of 1" in the optic angle'
Figure 11 is of similar form, but the variation is simpler in character.
The curves are valid for any value of B, and for both positive and negative substances.In contrast with Figs. 9 and 10, dB is smaller for small v
MAXIMUM
ERROR IN MINERAINGIC
COMPUTATIONS
691
values than for large. The overall range of dB is also considerablylessthan
for da and d7.
rf the refractive indices are determined by precision methods, dn will
be much less than .001. However, as the immersion method is rapid and
widely used, the diagrams have been constructedfor dn:.001.
CoupenarrvE AccuRACy.It has already beenpointed out that the minimum deviation method is capable of greater precision than any other.
rrowever, the small spectrometers used for this method in most laboratories give results which are usually less accurate than those obtained
with the Pulfrich refractometer, although adequate for practically all
mineralogic work.
In contrast, the extrapolation and optic angle methods, insofar as they
depend on immersion in oils, are much lessaccurate, giving third decimal
place accuracy only.
DnrpnurwarroN oF BrRonrrNcprqcn
GBunnar coMpENSAroRMETHoD(Bereh). Determination of refractive
indices to *.001 (immersion method) yields birefringence values accurate to + .002. This direct method may be compared with an indirect
method which makes use of the Berek compensator. Birefringence is determined by:
I/
t:
v**
where I': retardation in the unknown for a particular orientation.
t:thickness of thin section.
0/:angle between path of light and normal to the thin section.
By difierentiation, dB /B: dl'/ I/ - tan 0,d 0,- d,t/t. The thickness
t is found from a reference mineral of constant B (e.g. quartz), and
t: I" cos0" f B, where I" and 0" are analogousto I/ and 0/. Difierentiating, dt/t: df." f l" -tan 0,,d 0,,. By substitution in the original dB/B
equation, and assuming positive values for each term (to give the maximum deviation)
dB
dr,
dr,,
+ t a no ' do ' +
* t a no " d o " .
B-: F
r*
A generalexpressionfor dl'fl, and,dl,,f l,' may be obtained from the
basic compensatorformula, log l:log C*log f (i) where C:constant
of the compensator;ef(i):1un.,ion of angle of rotation i of compensator
drum. As f(i):sitrzi+.204 sinaif ...for a calcite compensator(Berek,
1913), differentiation of the compensator formula gives:
0 This constant must be determined for
each instrument.
H. W. FAIRBAIRN AND C. W. SHEPPARD
dr
sin 2i(1 + .408 sin'?i)di
sin2i * .204 sinai
The maximum expected uncertainty of a single reading of the compensatordrum is assumedto be 0.20.The uncertainty in 2i would then be
0.4" and for i would be 0.20.Then di:0.2/57.3:'00349 radians. Curve
df/f in Fig. 12 shows the per cent variation in fractional deviation with
changein I.r0 It is seenthat accuracy is much increasedwhere I is not too
small.
f (mf'
5oo
""
looo
1500
2000
Frc. 12. Berek compensator.Yariation in dI, and /6 accuracy of l., for difierent values
of I. Based on an uncertainty of * .2" in reading the compensator drum'
The uncertainties in 0' and 0" may be assumedequivalent. T'heir magnitude dependson (1) error made in orienting the sections,(2) error in
obtaining 0' and.0" on the stereographicnet, (3) error in the final hemispherecorrection. Error 3 applies only to the unknown, as the refractive
indices of the referencemineral are known. For the worst case,therefore,
a maximum uncertainty of !2" is not improbable. Where 0/ and 0" are
both small, however, errors 2 and 3 will be negligible and the uncertainty should not exceed t 1o.The generalimportance of small I values is
indicated clearly in Fig. 13.
No simple diagram will show the variation in accuracy of B, but the
total fractional deviation in any given case can be found by adding the
values of the four terms in the equation for dB/B.
r0 Thc values in Fig. 10 are about double those given by Berek (7924,Fig' 22,p'46)'
His calcul,ation,however, is of a "mean" error, not a maximum expected deviation'
MAXIMUM
ERROR IN MINERAINGIC
COMPUTATIONS
The preceding discussion implies strict uniformity in thin section
thickness. As this condition is seldom realized.,and, as it is physically
impossible for the reference mineral and the unknown to occupy the
same area of the section, a systematic error is thereby introduced which
has not yet been considered.In the caseof a strongly wedge-shapedsection, detection by means of interference colors is easy, and the mount
discarded if necessary.The thickness of apparently uniform sectionscan
be rapidly checked by orienting three or four widely separated reference
grains and noting the discrepancy in thickness obtained. Where this is
small, e.g., .002 mm. across the whole section, it can be shown that, if
unknown and referencemineral lie reasonably near to each other, a negligible error in B will result. In the case of mineral fragment mounts a
reference mineral can always be added, so that close spatial association
with the unknown is realized without difficulty.
In summary the accuracy of the indirect method increaseswhere (1)
thick rather than thin sectionsare used, (2) measurementsare made on
grains requiring the Ieast angular rotation for their orientation, (3) the
referencegrain is as closeas possibleto the unknown.
EulroNs ExrRApoLATroNMETHoD.A new orocedure described bv Emmons (1943, plate 13), based on the Biot-Neumann equation, permits
determination of total birefringence in any crystal where the optic plane
is vertical. The compensator used may be of fixed retardation (standard
gypsum or mica plate) or variable (Berek, graduated wedge, etc.). As
the analytical work on which Emmons' plate 13 is based has not been
published, no diagram of maximum error appears in this paper. Comment on the probable accuracy of the method will be found in the following section.
Oprrc eNcrr-coMpENSAroR MErHoD. The approximate relation
D-Bp
"t-
,h o .in /
developedby Biot and Neumann (Johannsen1918, p. 351), where Br
:total birefringenceof the crystal, Bp:partial birefringence(as measured on a randomly oriented crystal section), 0 and,0tare the angles subtended by each optic axis with the normal to the crystal section, becomes
81:(Bp/sin2 V) where the optic plane is parallel to the microscope
axis, with X or Z vertical, thus making 0:Y:0'.
A graphical solution
of this equation is given by Emmons (1943, plate 11). Where Br)0.1
there will be an appreciable correction to apply to V (see Larsen and
Berman, 1934,Fig. 1, or Emmons, 1943,plate 13). By difierentiation,
694
H. W. FAIRBAIRN AND C. W. SEEPPARD
dBr
dBp
Br:
B"
-2cotvdV'
dBp/Bp consists of four terms as described for the general compensator
method (Figs. 12 and 13). The additional term 2cotVdV is shown graphically in Fig. 13 for two values of dV. The uncertainty in measuring an
optic axis position need not exceed *1o; therefore if both axes can be
measured directly, the error in 2V is * 20 and in V is + 10. If, however,
only one axis position can be measured,the uncertainty in V is * 20.Both
these casesare included in Fig. 13. It is obvious that 2cotVdV is least
where V is large and that the method fails badly for small V values.
2C
Frc. 13. Tri'gonometric terms useil in the compensator melhotls. Tan 0d0 curve (general
compensator method) based on an error of *2o in 0. 2 cot VdV curves (optic angle-compensator method) based on an uncertainty in V of I 1oin curve (a) and of * 2' in curve (b).
CoupenauvE ACcuRAcy.Inspection of the differential equations derived for the first two birefringenie proceduresindicates that the general
method is potentially superior to the optic angle method. Presence of
the 2cotVdV term in the latter decreasesthe relative accuracy markedly. Total birefringence may be more accurately determined therefore
on an optic normal section using the general method.
It is probable that the fractional deviations in the Emmons extrapolation method are of the same order of magnitude as those in the optic angle method, i.e., somewhat less accuracy is to be expected than in the
general method.
Comparison of the general method (indirect) with direct determina-
MAXIMUM
ERROR IN MINERAI,OGIC COMPUTATIONS
695
tions of B may be made in an approximate way as follows. Where refractive indices are determined to *.001 (immersion method) the uncertainty in birefringencewould be *.002. For B:.01 the fractional
deviation is then 20/6;for B:.10 itis 2/s.Inspection of the range of
values in Figs. 12 and 13 for the indirect general method indicates that
a total fractional deviation in birefringence of I5/o might commonly
occur. By using thick sections and selecting grains which do not require appreciable rotation for their orientation, the above value might
easily be reduced to 5/s as a maximum expected error. Comparison of
the two methods depends therefore on auxiliary data, particularly on the
magnitude of B and d. For low birefringence the indirect general method
may, in many cases, be preferable; for high birefringence the direct
method is superior. Values of B obtained from precise prism determination of refractive indices would, of course, be superior in all casesto the
indirect method.
DnrrnurNarroN
or, Oprrc Axcrn
Accurate information regarding the size of the optic angle in biaxial
crystals is less important than accurate determination of refractive indices. However, as a number of proceduresfor optic angle determination
are in common use, it is advisable to consider their relative merits.
CoupnNsaroR METHoD.The approximate relation between optic angle
and birefringence, sinz V: B"rrBr is equivalent to sin 2 V: Ip/lr, where
retardations are computed for a common thickness of section, Ir is the
retardation in a section normal to X or Z, and Ir is the retardation in a
section normal to Y. Wright's graphical solution of this equation is given
by Emmons (1943,plate 11). Differentiating,ll
dV:
Irdlr
- fpdlr
I21 sin 2V
Using the Berek compensator, dlp and dlr are obtained from Fig. 12.
In Fig. 14 dV/V (%) is plotted against Ir for several selectedvalues of
V. In general, accuracy is seen to be greatest for large retardations; in
particular, where V is large as well.
As the basic formula above is an approximation, an appreciable correction to V must be applied where B)0.1 (Larsen and Berman,1934,
Fig. 1). This correction, however, would result in negligible change in
Fig. 14.
RnnnactrvB rNDEX METHoD. The approximate equations cos? V"
: @-o)/(t-o) : sin2V", and sin2Vo: 6-il /Q-d:
coS2Vz may be used
signin the expression
is changed
to positivefor computation
of maxi-"1lX:;.*"tive
H. W, FAIRBAIRN AND C. W. SHEPPARD
696
to determine V if all three indices are known. The subscript a refers to a
negative, 7 to a positive mineral. Difierentiating cos2Vd'
- B)dT
6 - ")dA + G - 7)da * ("
(t - o)'sin 2V
dV:
dV/V=15atr=200
Frc. 14. Cornpensotor methoil. Variation in accuracy oI V for difierent values of total
retardation (f1). Diagram shows curves for four values of V.
As a (g (?, dV will have its maximum value where dB is positive, da and
d7 are negative. The equation is then rewritten
dV:
(r -
Furthermore, da:dB:d7,
dV (radians) :
")d0
- @ - t)do - (o - ildt
(t - o)'sin 2V
permitting the substitution of dn, so that
2dn
or
(t
o)'sin 2V
dV (degrees) :
5 7 . 3X 2 d n
(t - o)'sin 2V
The uncertainty in n for the immersion method need not exceed
+.001, and for preciseprism work +.0001. Figure 15 is constructedfor
these values of dn, the curves representing various values of V as
marked. In general, accuracy is greatest for high birefringence; in particular for high V values. It is obvious that an optic angle computation
where dn: .001 is useful for relatively few substances(see Mertie, 1942),
and that prism refractometry is required if any precision in V is expected
by this method. Even with precision methods the accuracy is low if v is
small.
MAXIMUM
ERROR IN MINERAI.OGIC COMPUTATIONS
697
ExtrNcuou ANGLE MErHoD. This method, first made practical by
Berek (1924) and later amplified by Dodge (1934), provides a means of
determining the optic angle independently of either refractive indices or
retardation. It is discussedin two parts as outlined below.
Opti.c plane horizontal. This is the most useful case. The geometrical
relations are as follows (Dodge, 1934):
K+K'
Q:----;-r
tanK:
tan(O+ V) cosx,tanK/:
tan(S - V)cosx'
L
o
o
'i
c
!t
lic. 15. ReJraclile ind.er mel,hoil.Yaiation in accuracy of V for difierent values of total
birefringence (ry-a). Curves shown for optic angle values of 10", 20o, 30", 45'. Based
on refractive index uncertainties of .001 and .0001.
where the following rotations are made, in the order listedS is a fixed angle of rotation on the outer vertical axis of the stage;
x is a fixed angle of rotation on the outer east-westaxis of the stage;
@is an angle of rotation to extinction made on the microscopeaxis.
The graphical solution of these equationswill be found in Dodge (1934)
and also in Emmons (1943).
For the so-called "normal" case, where 6:45", and x is set equal to
some given angle, we assume V and 6 to be the only variablesl2in the
above relation. Differentiating,
dV:
2d6
sin 4d tan 2V
12The uncertainty in x arising from incorrect orientation of the optic symmetry plane
is lessened very considerably here by the large { value used, and is neglected in the difierential equation. Where { is small x cannot be considered constant.
698
H. W, FAIRBAIRN AND C. W, SHEPPARD
Fig. 16, showing dV/V plotted against V, contains two curves (full lines)
derived from this equation, for values of x:54.7o and 70o.The uncertainty in f need not exceed + 10 if the optic axes are more nearly horizontal than vertical for the orientation in which d is determined. If, however, the opposite condition holds, as it will in many cases,the uncertainty in @is likely to be more nearly * 20. As Fig. 16 is based on this
larger value, dV /V may be halved if the optic axes approximate a horizontal position under the conditions just stated. It is seenthat high V and
high x increasethe potential accuracy of the measurement. Two incom-
I
X = 5 47 '
X=7C
*r
={g
t0
\=20'-\
\-
,-N{ '
.a--
D"\
\
\_
__Ar
40
A5
Frc. 16. Erl,inction angl,emz,thoil (opl,ic plane horizontal). Variation in accuracy of Vfor
difierent values of V. FulI lines for 6:45' and x:54.7 and 70" as marked. Broken lines
and 30o. These four curves based on an uncertainty in @ of *2".
for x:70o and {:15"
Curves Dr and Dg, for direct readings of optic angle, are based on uncertainties in the optic
angle of 1o and 2o, respectively.
plete curvesfor [:15o and 30o (both for x:70o) have also beenplotted,
using the slopesof the proper curvesgiven by Dodge (1934,plates I and
II). Although these curves indicate higher accuracy than where 6:45o, it
must be recalled that x is not constant for such small values of n (see
footnote to previous paragraph). As the difierential equation for {(45"
and x variable is extremely cumbersome, no computations have been
made. However, the discrepancy in fractional deviation between curves
basedon 6:45" and {: 15ois lessthan Fig. 16 indicates.
Figure 16 includes also two curves, marked Dr and D2, which show the
variation in accuracy where V is measured directly. D1 is based on the
casewhere both optic axes are measuredand the uncertainty may attain
MAXIMUM ERRORIN MINERAI,OGICCOMPUTATIONS
699
a value of * 10;Dr is for the caseof one measuredaxis and an uncertainty
of *2". Assuming an uncertainty oI 20 in @,and d:45o, it is seenthat
the computed values of V approximate the accuracy of the direct method
only for the highest values of V; for small optic angles a small value of {
is required to attain comparable accuracy.
Optic Plane Verti,cal.This orientation is considered here only for the
caseof a horizontal acute bisectrix. Where the acute bisectrix is vertical
Frc. 17. Exlinction angle m,ethoil (optic ltlane terlical). Similar to Fig. 16.
it is usually possible to measure both axes directly and thus avoid indirect methods. The basic equation (Dodge, 1934)is:
/
secV:Isin:frf
\
c o s2 6
_
s r n "x
c o sx
\r/2
. " sin2gcot2gl
s t n "x
/
where {, x, and @are the same quantities used in the previous section and
rotations are made in the same order. As before, { and V are assumed
to be the only variables, although this is true only where x is not appreciably affected by incorrect setting of the optic symmetry plane. Differentiating,
dv-- -
cos x sin 26 cos Vdd
/
sin2x sin22g tan V ( sin2 6*
\
cos 26 cos x
\ 1/2
-sin 2$ cot 2S I
_
sln' x
sln" x
/
For selectedvalues of { and x this expressionbecomesmuch lessformidable in appearance. The uncertainty in @is taken as * 20, but with the
700
H. W. FAIRBAIRN AND C. W. SHEPPARD
same reservation as discussedin the previous section. Figure 17 shows
the variation in dV/V (7o) with change in V for two typical curves taken
from Dodge (1934,plates IV and V). In addition, a portion of the fractional deviation curve for the "normal" caseis included (upper right of
Fig. 17). Where the uncertainties in { are the same it is seenthat greater
accuracy can be obtained in the computation of V with small { than is
otherwise the case.13
The curves Dr and Dz described in Fig. 16 are replotted in Fig. 17 for comparison of direct and computed values of V.
As they cover about the same field in the diagram as the two curves
d:10o and {:15o, it is seen that there is little to choosebetween the
two methods under these conditions. If, however, the orientation conditions indicate a lower value of d{ than that on which Fig. 17 is based, the
computed value may be more accurate than that obtained by direct
measurement. In all cases,direct measurement is preferable to the computed value based on 6: 45".
INrnnnnnBNcE lrcuRE METHoDS.A number of methods based on interference figures are known (Johannsen 1918), but few are in widespread use at present. Where 2V<60" and a centered acute bisectrix
figure is obtainable, Mallard's method is convenient to use. The basic
relation is sin V:D/Kp,
where D:one-half the diptancebetween the
points of emergenceof the optic axes of the crystal plate, K:a constant
of the optical system (previously determined using minerals of known
optic angle), 0:intermediate refractive index of the crystal under examination.
Assuming any random error in K to be negligible relative to the uncertainty in measuring 2D, difierentiation of the above equation gives
If K:30 for the given optical system the uncertainty in 2D need not
exceed0.5 scaledivisions, so that dD:.25. If d0 is known to *.001, it
is easily shown that dB/B is negligible relative to dD/D, so that, finally,
dV (degrees):57.3dD/D tan V. Curve M in Fig. 18 is based on this
equation. It is seenthat the accuracy of the method is low for small optic
angles.
Curve C shows the variation of dV/V with V where the optic angle is
estimated from curvature of a single isogyre. With practice the uncertainty in estimating 2V by this methodra need not exceed 60, so that
dV:3". Curve C is basedon this value.
13The same comment made in the previous section regarding uncertainty in x for low
n
values applies here.
la E. S. Larsen, personal communication.
MAXIMUM
ERROR IN MINERALOGIC COMPUTATIONS
701
For comparison, curve Dr (showing the fractional deviation in the direct method for dV:1o) is included in Fig. 18.
CoupanerrvE ACcuRACy.Inspection of the diagrams (Figs. 14-18)
makes apparent the difficulty of general comparison of optic angle methods. Except where V and B are smail, the refractive index method is
$z
l5'
300
a5n
Frc. 18. Inlerlerence f.gule methods. Variation in accuracy of V for Mallard's method
(curve M), where K:30 and dD:.25. Curve C represents the estimation method where
dV:3'. Curve Dr is for the direct method where dV:1'.
undoubtedly the most accurate if the indices are determined by the
minimum deviation prism method. However, where the indices are determined by matching with an oil to *.001, the method yields results
which are not much better on the average than results obtained from
estimating the curvature of an isogyre.
Where favorable conditions for rotation exist, and with the uncertainty in S not in excessof 1o, the extinction angle method will give
values of V almost as free from error as the refractive index method (for
dn:.0001 and B>.02). Where the uncertainty is *20, however, the
accuracy is approximately the same as the refractive index method with
dn:.001.
Comparison of the compensator method with other methods requires
the assumption of a definite thickness for the section. Thus for t:.02
mm. the dV/V compensatorvalues are about intermediate between those
obtained from the extinction ansle and refractive index methods.
702
H. W, FAIRBAIRN
AND C. W. SEEPPARD
Although of restricted application, Mallard's interference figure
method is more accurate than direct measurement of optic angle with
the universal stage, if the maximum values for dD and dV used in the
diagrams are accepted as probable.
Comparison of the three indirect universal stage methods with the
direct method shows in most casesthat the latter is at least equal, or
definitely superior, in accuracy to the former. Exceptions occur with the
refractive index method where dn:.0001 and B and V are moderate to
Iarge,and with the extinction angle method where fr is small and x large.
For all methods less accuracy is possible for low values of optic angle
than for high values.
Suuueny
The usefulnessof the diagrams presented in this paper depends on
good judgment in selecting a reasonable maximum value of the uncertainty in any given measurement.In choosingthe values on which the
preceding figures are based the authors have assumeda skilled operator
who has at his disposal good material and equipment. Where a seriesof
readings of the same measurementis taken and a "probable error" computed, greater accuracy should be attained than shown here. Where the
working material is poor less accuracy is to be expected. In general,
where the uncertainties in measurement are believed to differ from those
used in the diagrams, it is necessaryonly to substitute in the differential
equation the revised value for the original and to construct a new graph.
A set of these graphs readily available in the laboratory should prove
highly useful by providing instant information regarding the order of
magnitude of the maximum error to be expected in a given procedure. It
is the authors' hope that in time a more realistic attitude will be developed toward mineralogic measurementsthan is now apparent from much
of the published quantitative data.
Ac<wowrrnGMENTS
The writers are indebted to Dr. E. S. Larsen of llarvard lJniversity
for critical examination of the manuscript and for several helpful suggestions.
Rnrnnnucns
BANNrstnn,F. A., llrn Hrv, M. H. (1937), A new micro-pyknometric
methodetc.:
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